Functional Analysis is a subject that combines Linear Algebra with Analysis. I researched online, and it seems one of the best Functional Analysis Book for Graduate level is Functional Analysis by Peter Lax. This book is ideal for a second course in Functional Analysis. For a first course in Functional Analysis, I would recommend Kreyszig, which is listed on my Recommended Undergraduate Math Books page.
This is Theorem 5 in the book: Let be a linear space over the reals.
The following 9 properties hold:
- The empty set is convex.
- A subset consisting of a single point is convex.
- Every linear subspace of is convex.
- The sum of two convex subsets is convex.
- If is convex, so is .
- The intersection of an arbitrary collection of convex sets is convex.
- Let be a collection of convex subsets that is totally ordered by inclusion. Then their union is convex.
- The image of a convex set under a linear map is convex.
- The inverse image of a convex set under a linear map is convex.
Brief sketch of proofs:
We give a brief sketch of the idea behind the proofs.
We are using the definition of convex as follows: is a linear space over the reals; a subset of is called convex if, whenever and belong to , all points of the form , also belong to .
Property 1 is vacuously true.
Property 2 is true because of .
Property 3 is true because is a linear combination and is thus in the linear subspace.
Property 4) Let and be the two convex subsets. Let and be points in
Property 5) We just need to know that and this algebraic observation: .
Property 6) Let . for all , thus .
Property 7) Let , where . Let , , then either or . If , . Similarly for the other case .
Property 8) Observe that .
Property 9) The only tricky thing about this part is that we cannot assume that the inverse exists. We can only talk about the pre-image.
Let . and .
We have .